Bielliptic ball quotient compactifications and lattices in \(\operatorname{PU}(2,1)\) with finitely generated commutator subgroup (Q1687865)

A complete complex hyperbolic surface, or a ball quotient surface, is a complete Hermitian surface of constant holomorphic sectional curvature \(-1\). The purpose of this paper is to produce two infinite families of explicit bielliptic ball quotient compactifications. Theorem 1.1. For any natural number \(n,\) there exists a smooth projective surface \(X_{n}\) birational to a bielliptic surface and neat lattice \(\Gamma_{n}\) in \(\mathrm{PU}(2,1)\) of covolume \[ \frac{8}{3} \pi^{2} n \] such that \(X_{n}\) is a smooth toroidal compactification of \(B^{2}/\Gamma_{n}.\) In particular, the family \(B^{2}/\Gamma_{n}\) saturates the entire admissible volume spectrum of ball quotient surfaces with holomorphic sectional curvature \(-1\).